Pretty much any study of the past century and any projection from reputable sources says that populations reach equilibrium and stop growing as the level of education, in particular education of women, rises.
This is a non-problem that has always taken care of itself in any developed country and we have no reason to believe it will not take care of itself in the developing world as well.
The UN for instance does not believe there will be 10 billion humans on earth ever (where "ever" means "as long as projections have any value").
I wouldn't quite call world population growth a non-problem, considering it is already a problem. The future predictions may well turn out correct, and have some solid reasoning behind them, but the numbers still look fairly crude to me. Plots of historic growth rates are bouncing up and down, while the prediction is they will suddenly turn a corner tomorrow and drop monotonically to zero and all will be peachy. How often has that happened? Considering the current record-breaking population numbers, the ultimate answer is it has never happened, at least not permanently. There's a lot of risk in betting on things to just work themselves out. Development can actually work against us in a sense, as new food production technology and cures allow faster-than-ever growth in new regions and populations.
Below replacement fertility is a recent phenomenon (last couple of generations). Evolution works in multiples of generations. We’re at the very beginning of evolving resistance to modernity.
This is a non-problem that has always taken care of itself in any developed country and we have no reason to believe it will not take care of itself in the developing world as well.
The UN for instance does not believe there will be 10 billion humans on earth ever (where "ever" means "as long as projections have any value").